As of December 2017, the largest known prime number is 2 77,232,917 − 1, a number with 23,249,425 digits. It was found on December 26, 2017 by the Great Internet Mersenne Prime Search (GIMPS).

Euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.

Many of the largest known primes are Mersenne primes. As of January 2017, the six largest known primes are Mersenne primes. [2] The last 16 record primes were Mersenne primes. [3] [3]

The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers.

The current record

The record is currently held by 2 77,232,917 − 1 with 23,249,425 digits. [38]

Prizes

The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits.

There are several prizes offered by the Electronic Frontier Foundation for record primes. [21] GIMPS is also coordinating its long-range search efforts for primes of 100 million digits and larger and will split the Electronic Frontier Foundation's US$150,000 prize with a winning participant.

The record passed one million digits in 1999, earning a US$50,000 prize. [13] In 2008 the record passed ten million digits, earning a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation. [21] Time called it the 29th top invention of 2008. [8] Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits. [21] Both the US$50,000 and the US$100,000 prizes were won by participation in GIMPS.

History

The following table lists the progression of the largest known prime number in ascending order. [3] Here M n = 2 n − 1 is the Mersenne number with exponent n . The longest record-holder known was M 19 = 524,287, which was the largest known prime for 144 years. Almost no records are known before 1456.

Number Decimal expansion
(only for numbers < 10 50 )
Digits Year found Notes
(for larger Mersenne primes, see Mersenne prime)
11 11 2 ~1650 BCE ancient Egyptians (disputed) [14]
7 7 1 ~400 BCE It was known to Philolaus that 7 is a prime [15]
M 7 127 3 ~300 BCE It was known to Euclid that 127 and 89 are primes [2] [17]
M 13 8,191 4 1456 Anonymous discovery
M 17 131,071 6 1460 Anonymous discovery
M 19 524,287 6 1588 Found by Pietro Cataldi
6,700,417 7 1732 Found by Leonhard Euler
M 31 2,147,483,647 10 1772 Found by Leonhard Euler
67,280,421,310,721 14 1855 Found by Thomas Clausen
M 127 170,141,183,460,469,231,731,687,303,715,884,105,727 39 1876 Found by Édouard Lucas
20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 44 1951 Found by Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M 127 ) 2 +1 79 1951 Using Cambridge's EDSAC computer
M 521 157 1952
M 607 183 1952
M 1279 386 1952
M 2203 664 1952
M 2281 687 1952
M 3217 969 1957
M 4423 1,332 1961
M 9689 2,917 1963
M 9941 2,993 1963
M 11213 3,376 1963
M 19937 6,002 1971
M 21701 6,533 1978
M 23209 6,987 1979
M 44497 13,395 1979
M 86243 25,962 1982
M 132049 39,751 1983
M 216091 65,050 1985
391581×2 216193 −1 65,087 1989
M 756839 227,832 1992
M 859433 258,716 1994
M 1257787 378,632 1996
M 1398269 420,921 1996
M 2976221 895,932 1997
M 3021377 909,526 1998
M 6972593 2,098,960 1999
M 13466917 4,053,946 2001
M 20996011 6,320,430 2003
M 24036583 7,235,733 2004
M 25964951 7,816,230 2005
M 30402457 9,152,052 2005
M 32582657 9,808,358 2006
M 43112609 12,978,189 2008
M 57885161 17,425,170 2013
M 74207281 22,338,618 2016

GIMPS found the thirteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.

The twenty largest known prime numbers

A list of the 5,000 largest known primes is maintained by Chris K. Caldwell, [2] the twenty largest are listed below.

Rank Number Discovered Digits Ref
1 2 74,207,281 – 1 2016-01-07 22,338,618
2 2 57,885,161 – 1 2013-01-25 17,425,170 [19]
3 2 43,112,609 – 1 2008-08-23 12,978,189 [20]
4 2 42,643,801 – 1 2009-06-04 12,837,064 [21]
5 2 37,156,667 – 1 2008-09-06 11,185,272 [20]
6 2 32,582,657 – 1 2006-09-04 9,808,358 [22]
7 10223 ×  2 31,172,165 + 1 2016-10-31 9,383,761 [23]
8 2 30,402,457 – 1 2005-12-15 9,152,052 [24]
9 2 25,964,951 – 1 2005-02-18 7,816,230 [25]
10 2 24,036,583 – 1 2004-05-15 7,235,733 [26]
11 2 20,996,011 – 1 2003-11-17 6,320,430 [27]
12 919,444 1,048,576 + 1 2017-08-29 6,253,210 [28]
13 168,451 × 2 19,375,200 + 1 2017-09-17 5,832,522 [29]
14 123,447 1,048,576 − 123,447 524,288 + 1 2017-02 5,338,805 [30]
15 143,332 786,432 − 143,332 393,216 + 1 2017-01 4,055,114 [31]
16 2 13,466,917 − 1 2001-11-14 4,053,946 [32]
17 19249  ×  2 13,018,586 + 1 2007-05 3,918,990 [34]
18 3  ×  2 11,895,718 − 1 2015-06-23 3,580,969 [35]
19 3  ×   2 11,731,850 − 1 2015-03-13 3,531,640 [36]
20 3  ×   2 11,484,018 − 1 2014-11-22 3,457,035 [37]

See also